The Study On The Evolving Model Of Complex Networks And Their Fractality

In the real world, there are many systems, which can be represented by complex networks, such as Internet, movie actor collaboration network, metabolic networks, aviation network, and so on. In the complex network, the nodes represent individuals or organizations and edges mimic the interactions among them. And many empirical studies indicate that these various kinds of networks have some common characteristics: power-law distribution of degree, high clustering coefficient, and small average path length. These characteristics receive the researchers' great interest and considerable attention. This thesis takes mathematic analytical method, fractal, and computer simulation, these methods into the complex network, and builds some improved models on the complex network to study on the character of the complex network.This thesis first introduces the background and the significance of the complex networks and the relevant conceptions are explained, too. Then to solve the limitation of the models which have been introduced, three new better models are proposed:1. A scale-free network model with the changing scaling exponent is proposed. The degree distribution in many real networks decays as a power law, and the scaling exponent is scattered between 2 and 3, while the exponent of BA model is 3 constantly. Based on the BA model, we import the preferential attachment mechanism between the old vertexes and propose a scale-free model in which the sealing exponent can change in some certain scope. The paper calculates analytically the degree distribution and the result indicates that the scaling exponent varies 2 to 3 under different values of the parameters. It is also pointed that the BA model is a particular case of the model. And through the analysis of some data of the real networks, the model is proved to be reasonable and effective.2. A model of Apollonian Networks with aging mechanism is proposed. This paper proposes a model of Apollonian Networks with aging mechanism based on the former models on Apollonian Networks. The analytical result and analog computation is proved that the model is small world and power-law. Also, we analytically calculate the distribution of age of nodes, and prove that there is a negative correlation between the age of a node and the probability of its getting new link. Therefore, the model proposed in this paper performs better than Random Apollonian Networks. And it is of more effective practicability for layout of planar networks, such as printed circuits.3. A class of graphs with fractality and small-world effect is proposed. The clustering coefficient, the average length of the graphs, and the diameter are calculated analytically. Also we calculate the box-counting dimension and Hausdorff dimension as a measure of their fractality and the value of the dimension is 1.585. At last, the method of the constructing graphs is expanded, and the expressions of the relevant characteristics of the expanded graphs are given. The expanded graphs and the graphs proposed before are considered as a class of graphs with fractality and small-world effect.